Non-stationary characteristics and structural optimization of CSR section under open-water condition

Abstract

To achieve energy conservation and improve ship maneuverability, a new type of collaborative spoiled rudder (CSR) is proposed by absorbing the advantages of existing ship rudders. This study investigates the non-stationary characteristics of the hydrodynamic forces of CSR section in depth, and carries out structural optimization based on numerical simulations. First, the effects of rudder angle and spoiler open angles on the non-stationary characteristics of the hydrodynamic forces of the CSR, and the appropriate maximum value of the spoiler open angle is determined. The results show that the unsteady pulsation amplitude of the hydrodynamic forces of the CSR is directly proportional to the spoiler open angle and inversely proportional to the rudder angle, and the upper limit of the spoiler open angle is set to 30°. Then, the hydrodynamic characteristics of the CSR with different spoiler widths are investigated in further. The maximum lift of the CSR increases as the spoiler width increases. The target lift method is used for efficiency evaluation. The ranges of the favorable rudder angle and the favorable spoiler open angle for different CSR configurations are determined, and the optimal spoiler width is further determined based on the non-stationary characteristics of the hydrodynamic forces.

Appendix

Mesh uncertainty analysis about the NACA0012 section was carried out, and the results (lift and drag coefficients) were compared with the corresponding experimental data to validate the mesh generating strategy and numerical method.

The three-mesh estimate method was adopted for the mesh uncertainty analysis, and three meshes (coarse mesh G01, median mesh G02 and fine mesh G03) were produced with the refinement factor of $\sqrt 2$. The C_L and C_D values were used as variables. G01 and G03 were obtained by increasing or decreasing the grid size by a factor of $\sqrt 2$ globally based on G02. The grid size of the median mesh G02 was dynamically adjusted until the results did not change significantly with a finer mesh.

Table 4 presented the final results of the mesh uncertainty analysis. The specific calculations of the parameters given in Table 4 were based on the work of Stern et al. [20]. For the median mesh G02, the chord-spacing of the mesh along with the section was 1.5 × 10^–3 C, and the layer-wise growth rate of the boundary layer was 1.1, as proposed by Liu et al. [16]. The first layer thickness of the medium mesh was set to 2.2 × 10^–6 C, corresponding to a y+ of 0.5. A y+ of this size could be sufficient to properly resolve the inner parts of the boundary layer. The coarse G01, median G02, and fine G03 meshes had 0.98 × 10⁵, 1.4 × 10⁵ and 2.1 × 10⁵ cells, respectively. The corresponding time step for different mesh structures was determined under the condition that the target max CFL was 5.0 and the mean value was 0.5. The results demonstrated that the hydrodynamic forces decreased with the increase in cell number under a monotonic convergence with a convergence ratio 0 < R_G < 1. This indicated that the impact of the mesh change was accepted to be small between G02 and G03. Besides, the reasonable small levels of the mesh uncertainty (${\text{U}}_{\text{G}}$) for the fine mesh set G01 were estimated. The results demonstrated that the satisfactory mesh uncertainties were obtained.

Table 5 showed the lift and drag coefficients of median mesh G02 and the corresponding experimental data [21]. The simulated results exhibited a good agreement with the experimental data.

Table 4 Grid convergence study

Table 5 Calculation errors for median mesh